4.2 Trigonometry 2354.2.2 Euler’s Formula..........................................
For a complex numberz,
ez= 1 +z
1!+
z^2
2!+···+
zn
n!+···.
In particular, for an anglex,
eix= 1 +ix
1!−
x^2
2!−ix^3
3!+
x^4
4!+ix^5
5!−
x^6
6!−ix^7
7!+···.
The real part ofeixis
1 −
x^2
2!+
x^4
4!−
x^6
6!+···,
while the imaginary part is
x
1!−
x^3
3!+
x^5
5!−
x^7
7!+···.
These are the Taylor series of cosxand sinx. We obtain Euler’s formula
eix=cosx+isinx.Euler’s formula gives rise to one of the most beautiful identities in mathematics:eiπ=
−1, which relates the numberefrom real analysis, the imaginary unitifrom algebra,
andπfrom geometry.
The equalityenz=(ez)nholds at least forza real number. Two power series are
equal for all real numbers if and only if they are equal coefficient by coefficient (since
coefficients are computed using the derivatives at 0). So equality for real numbers means
equality for complex numbers. In particular,einx=(eix)n, from which we deduce the
de Moivre formula
cosnx+isinnx=(cosx+isinx)n.We present an application of the de Moivre formula that we found inExercises and
Problems in Algebraby C. Nast ̆ asescu, C. Ni ̧t ̆ ̆a, M. Brandiburu, and D. Joi ̧ta (Editura
Didactic ̆a ̧si Pedagogic ̆a, Bucharest, 1983).
Example.Prove the identity
(
n
0)
+
(
n
k)
+
(
n
2 k)
+··· =
2 n
k∑kj= 1cosn
jπ
kcos
nj π
k